A two-strain SARS-COV-2 model for Germany - Evidence from a Linearization - arXiv
←
→
Page content transcription
If your browser does not render page correctly, please read the page content below
A two–strain SARS–COV–2 model for Germany -
Evidence from a Linearization
Thomas Götza,∗, Wolfgang Bockb , Robert Rockenfellera , Moritz Schäfera
arXiv:2102.11333v1 [q-bio.PE] 22 Feb 2021
a Mathematical Institute, University Koblenz, 56070 Koblenz, Germany
b Department of Mathematics, TU Kaiserslautern, 67663 Kaiserslautern, Germany
Abstract
Currently, due to the COVID–19 pandemic the public life in most European
countries stopped almost completely due to measures against the spread of the
virus. Efforts to limit the number of new infections are threatened by the advent
of new variants of the SARS–COV–2 virus, most prominent the B.1.1.7 strain
with higher infectivity. In this article we consider a basic two–strain SIR model
to explain the spread of those variants in Germany on small time scales. For a
linearized version of the model we calculate relevant variables like the time of
minimal infections or the dynamics of the share of variants analytically. These
analytical approximations and numerical simulations are in a good agreement
to data reported by the Robert–Koch–Institute (RKI) in Germany.
Keywords: COVID–19, Epidemiology, Disease dynamics, Multi–strain model.
1. Introduction
The current COVID–19 pandemic is striking across the world and has put
Europe at the dawn of its third wave. In Germany due to the rising numbers at
the end of the year 2020, the non-pharmaceutical intervention (NPI) measures
have been strengthened, leading to a severe lockdown with closing of the main
parts of the daily life. With the reestablishment of the NPIs the reports of new
strains of the SARS–COV–2 virus throughout Europe were rising [1]. Especially
a variant called B.1.1.7 that was first reported in Great Britain [4, 2], showed
an increased infectivity[3] with a higher attack rate especially in the younger
age groups. In the last days (February 21, 2021) the incidences are stagnat-
ing or slowly increasing, although Germany has not eased the lockdown. One
explanation among experts and media is the rising of incidences with the new
mutations.
In various countries B.1.1.7. is rapidly spreading, see [5] for an overview. In
∗ Corresponding author.
Email addresses: goetz@uni-koblenz.de (Thomas Götz), bock@mathematik.uni-kl.de
(Wolfgang Bock), rrockenfeller@uni-koblenz.de (Robert Rockenfeller),
moritzschaefer@uni-koblenz.de (Moritz Schäfer)
1100
England
Denmark
Portugal
80 Netherlands
Germany
B.1.1.7 variant in %
60
40
20
0
0 2 4 6 8 10 12 14
Week after 1%
Figure 1: Share of the B.1.1.7 variant of the SARS–COV–2 virus in five European countries.
Week zero corresponds to the week when the share is approximately 1%.
Figure 1 we show the share of this new strain with respect to analyzed SARS–
COV–2–positive tests in a given week in five European countries. Week zero is
defined as the week, when this share was approximately 1%. All five countries
follow a general logistic trend. The curves for England (blue), Netherlands
(orange) and Germany (black) are rather similar, where as Denmark (red) and
Portugal (green) also behave similar but slower than the first three. Within
this paper we will formulate a SIR–based model that predicts these curves and
explains well the observed data in Germany on a small time horizon.
2. Mathematical Model
We consider an SIR–model for the spread of two strains of the SARS–COV–2
virus within a constant and serologically naive population. The two competing
strains 1 and 2 are assumed to have different transmission rates β2 , β1 > 0 but
the same recovery rate γ > 0. Assuming no secondary infections, the model is
based on the four compartments: susceptible S, infected I1 and I2 , indicating
strain 1 or 2,respectively, and removed R. Neglecting demographic effects like
2birth and death, we get
S
S ′ = − (β1 I1 + β2 I2 ) , (1a)
N
β1 I1
I1′ = S − γI1 , (1b)
N
β2 I2
I2′ = S − γI2 , (1c)
N
R′ = γ(I1 + I2 ) . (1d)
For the following analysis and simulations we assume a situation that models
the competition between the original SARS–COV–2 virus and mutated variants
like B.1.1.7 that is currently observed in many countries throughout Europe.
The second (mutated) strain has a higher infection rate, i.e. β2 > β1 . However,
at the initial time, the original strain 1 is still dominant in the population
i.e. I1 (0) > I2 (0). Current non–pharmaceutical interventions are strict enough
to suppress the original strain, i.e. to force its reproduction number below the
epidemic threshold R1 := βγ1 < 1. However the mutated strain 2, due to its
higher infectivity, might reach a reproduction number R2 > 1 and hence drives
the epidemic.
Typical questions that might arise in this setting:
• When will strain 2 dominate the dynamics? After what time T ∗ do we
observe I2 (T ∗ ) > I1 (T ∗ )?
• How does the total number of infected I = I1 + I2 evolve in time? At
what time Ť do we observe a local minimum of the infections?
In our model, we neglect the effect of possible vaccinations, that might have
different efficiency with respect to the two strains.
3. Analysis
Let N denote the constant total population. We rescale the populations
s = S/N , x = I1 /N , y = I2 /N and r = R/N and introduce a non–dimensional
time γt. Then we get
s′ = − (R1 x + R2 y) s (2a)
′
x = (R1 s − 1) x (2b)
′
y = (R2 s − 1) y (2c)
r′ = x + y (2d)
where Ri = βi /γ. Setting y = zx, where z denotes the ratio between infected
with strains 1 and 2, we get
z ′ = (R2 − R1 ) sz (3a)
3with the solution
Z t
z(t) = z0 exp (R2 − R1 ) s(t) dt . (3b)
0
In case of R2 > R1 , the ratio between the two strains is going to tend towards
strain 2, i.e. z > 1.
3.1. Linearized Setting
In case of dominating susceptibles, i.e. s ≈ 1 the ODEs (2) linearize
x′ = (R1 − 1) x (4a)
y ′ = (R2 − 1) y , (4b)
and we are able to solve them explicitly for the infected compartments x, y. For
both compartments we observe an exponential behavior; however since R1 < 1
the compartment x is dying out and compartment y is exponentially growing
due to R2 > 1. The ratio z of the two strains exhibits an exponential increase
z(t) = z0 e(R2 −R1 )t . (5)
In this setting we can easily answer the initial questions posed in section 2:
1. Strain 2 will ”overtake” strain 1 at time T ∗ , i.e. z(T ∗ ) = 1. In the
linearized model (4) this time T ∗ is given by
ln z0
T∗ = − >0 (6)
R2 − R1
since z0 < 1 and R2 > R1 .
2. The total infected attain a local minimum at time Ť when (x+ y)′ (Ť ) = 0.
In the linearized model (4) it holds that
x′ + y ′ = (R1 − 1)x + (R2 − 1)zx
1−R1
and hence z(Ť ) = z0 e(R2 −R1 )Ť = R2 −1 > 0. So we arrive at
1 1 − R1 1 − R1
Ť = ln = T ∗ · ln (7a)
R2 − R1 z0 (R2 − 1) R2 − 1
1 − R1
ž = z(Ť ) = . (7b)
R2 − 1
The minimal number of infected is given by
(x + y)min = x0 e(R1 −1)Ť (1 + ž)
(R1 −1)/(R2 −R1 )
1 − R1 R2 − R1
= x0 · (7c)
z0 (R2 − 1) R2 − 1
4The relative share p = y/(x + y) = z/(1 + z) of the second strain y with
respect to the total infected x + y satisfies in the linearized setting the following
logistic relation
z0 e(R2 −R1 )t z0
p= = ∈ [0, 1] . (8)
1 + z0 e(R2 −R1 )t z0 + e−(R2 −R1 )t
In the linear, single strain model x′ = (R1 − 1)x the reproduction number
satisfies the relation
d
R1 = 1 + ln x .
dt
Hence we may define analogously the current reproduction number R(t) for the
total infected as
d
R(t) := 1 + ln(x + y) . (9)
dt
Using the solution of the linear model x + y = x0 e(R1 −1)t + x0 z0 e(R2 −1)t we
obtain the convex combination
R2 − R1
R(t) := (1 − p)R1 + pR2 = R1 + 1 −(R2 −R1 )t , (10)
1+ z0 e
i.e. a logistic behavior switching between R1 for t → −∞ and R2 for t ≫ 1.
At time Ť , when the total number of infected attains its minimum, the current
reproduction number crosses the stability threshold, i.e. R(Ť ) = 1. For the
non–linear model, the overall behavior of the reproduction number is similar,
despite the saturation effect due to the decreasing pool of susceptibles.
4. Simulations
For our simulations, we assume the following data roughly resembling the
situation in Germany by mid of January to mid of February:
1. The total population equals to N = 83 millions including 3 millions of
recovered or vaccinated and x0 ≃ 78.000 infected with variant 1 and y0 =
z0 · x0 infected with strain 2.
2. The recovery period is assumed to be 1/γ = 5 days.
3. Strain 1 has a reproduction number of R1 = 0.85, i.e. the current lock-
down measures are strict enough to mitigate the original strain.
4. The mutated strain 2 is assumed to be 50% more infectious, i.e. R2 =
1.5 · R1 = 1.275 and hence spreads in time.
5. At the initial time (25 January) we assume that only 3% of cases belong
to strain 2, i.e. z0 = 0.03.
6. Lab experiments [9] report in week 4 around 5.6% and in week 6 already
around 22% of infections with strain 2.
5200
RKI-Data
175 Incidence 35
Strain 1
Strain 1+2
150 R2 ± 0.1
125
7 day incidence
100
75
50
25
0
11.Jan 25.Jan 08.Feb 22.Feb 08.Mar 22.Mar
Date
Figure 2: Incidence (per 100.000 inhabitants in 7 days) of strain 1 (violet) and of both strains
combined (blue) as predicted by the SIR–model (2). Parameters are R1 = 0.85, R2 = 1.275,
γ = 1/5. The green dots indicate incidences for entire Germany. The shaded area indicates
the simulation range, if R2 = 1.275 ± 0.1. The orange dash line indicates an incidence of 35
that is viewed in Germany as a limit for relaxing the current lockdown.
Using the approximations (6) and (7) from the linearized model, strain 2
will dominate strain 1 at T ∗ = − Rln z0
2 −R1
= 8.2 ≡ 41 days. The minimal number
of infected is expected to be (x + y)min = 0.69 · x0 at time Ť = 6.8 ≡ 34 days.
The non–linear model (2) cannot be solved analytically; hence we perform
numerical simulations based on the parameter given above. Figures 2, 3 show
the dynamics of both strains and the current reproduction number based on
Eqn. (9). The weekly incidences (new infections per 100.000 inhabitants within
7 days) are shown in Figure 2. The green dots indicate the reported data,
see. [10]. The violet curve shows the decay of the original strain 1 with is
reproduction number R1 = 0.85 < 1. The blue curve shows the total incidence
of both strains combined. At around 8 March, i.e. 6 weeks after the starting time
of the simulation (25 January), the total number of infected reaches its minimum
with an incidence of about 44 per 100.000. The time at which the minimum
occurs is slightly larger than for the linear model (41 days for the nonlinear
model compared to 34 days for the linearized approximation). Our scenario
and its parameters match quite well with the observed incidences, here shown
for 8 and 17 February. The shaded area indicates the prediction uncertainty
due to variations (R2 ± 0.1) of the reproduction number of the second strain.
Based on this simulation, the political target to push the infections below the
threshold of 35 before introducing relaxation measures seems questionable; at
least on a short time horizon.
6current R-value
R2 ± 0.1
1.2 RKI-data for R-interval
1.1
R(t)
1.0
0.9
0.8
01.Feb 15.Feb 01.Mar 15.Mar 29.Mar
Date
Figure 3: Current value of the effective reproduction number (9) for both strains. The green
dots indicate the RKI–data for the 7day–R0 together with the reported confidence interval.
The shaded area indicates the simulation range, if R2 = 1.275 ± 0.1.
Figure 3 shows the current reproduction number for both strains combined
as defined in Eqn. (9). Again, the green dots and error bars show the 7–day
reproduction number as reported by RKI in its daily situation reports [11].
The blue curve shows our simulation results based on the non–linear model (2)
and again the shaded area indicates the uncertainty due to variation of the
reproduction number of the second strain (R2 ±0.1). Currently, RKI is reporting
an increase of the reproduction number breaking through the epidemic threshold
of R(t) = 1 as predicted by our model. The non–pharmaceutical interventions
imposed by the government have not been altered in during the time span
covered by the simulations, hence we may explain the increase of the overall
reproduction number by growing influence of the second strain.
Figure 4 shows the relative share p of strain 2 with respect to the total
infections with SARS–COV–2 in Germany. The green dots indicate the data
reported by RKI for week 4 and week 6, see [9]. The blue curves show our pre-
dictions; the dashed one corresponds to the approximation (10) in the linearized
setting and the solid one corresponds to the non-linear SIR model. Both results
do not differ significantly and the linearized model already predicts quite well
the dynamics of the relative share of the second strain compared to all infec-
tions. Both results are within the range of the given data. Again, the shaded
area indicates the uncertainty caused by variations in the reproduction number
for the second strain. For the beginning of March we expect more than 40% of
infections with the second strain.
7100
SIR-model
linearized model
RKI Data
80 R2 ± 0.1
60
Strain 2 in %
40
20
0
01.Feb 15.Feb 01.Mar 15.Mar 29.Mar
Date
Figure 4: Simulation for of the relative share p = y/(x + y) of the second strain. Reproduction
numbers R1 = 0.85, R2 = 1.275. The green dots indicate the RKI data for the percentage of
”variants of concern”, see [9]. The solid blue line shows the non–linear SIR–Model (1d) whereas
the dashed line is the linearized approximation. The shaded area indicates the simulation
range of the non–linear SIR–model, if R2 = 1.275 ± 0.1.
5. Conclusions and Outlook
In this work we have presented a two–strain SIR model to explain the spread
of SARS–COV–2 variants like B.1.1.7 in Germany. For a linearized version of
the model we were able to calculate relevant variables like the time of minimal
infections or the dynamics of the share of variants analytically. These analyt-
ical approximations as well as simulations for the non–linear SIR model are
compared to infection data reported by RKI. Our model shows a good level of
agreement and gives rise to some concern regarding the near term future of the
dynamics. For mid of March we expect to see in Germany a share of at least
40% of variants. Moreover, the figure of an incidence of 35 per 100.000 and 7
days, which was introduced by politics as limit for easing the current lockdown
measures, seems out of reach.
In a follow–up study we will try to investigate the effect of the current
ramping up of mass vaccinations. One might expect, that vaccinations will help
to slow down the spread of the disease and hence force the level of incidence
below a threshold that allows contact tracing by public health authorities.
References
[1] Pachetti, M., Marini, B., Benedetti, F., Giudici, F., Mauro, E., Storici,
P., et.al. Emerging SARS–CoV–2 mutation hot spots include a novel RNA-
8dependent-RNA polymerase variant. Journal of translational medicine, 18,
1-9. 2020.
[2] Tang, J. W., Tambyah, P. A., & Hui, D. S. Emergence of a new SARS-
CoV-2 variant in the UK. The Journal of infection. 2020.
[3] Korber, B., Fischer, W. M., Gnanakaran, S., Yoon, H., Theiler, J., Ab-
falterer, W., et.al. Tracking changes in SARS–CoV–2 Spike: evidence that
D614G increases infectivity of the COVID–19 virus. Cell, 182(4), 812-827.
2020.
[4] Volz, E., Mishra, S. and Chand, M., Transmission of SARS–CoV–2 lineage
B. 1.1. 7 in England: insights from linking epidemiological and genetic
data. medRxiv [Preprint posted online January 4, 2021].
[5] Wikipedia: Development of the B.1.1.7.lineage
https://en.wikipedia.org/wiki/Lineage_B.1.1.7#Spread_in_Europe;
access on 21.February.
[6] Götz, T.; Heidrich, P.: Early stage COVID-19 disease dynamics in Ger-
many: models and parameter identification. J. Math. in Industry. No. 10,
(2020).
[7] Heidrich, P., Schäfer, M., Nikouei, M., Götz, T.: The COVID–19 outbreak
in Germany — Models and Parameter Estimation. Commun. Biomath. Sci.
3, 37-59, 2020.
[8] Sueddeutsche Zeitung: Die unsichtbare Welle
www.sueddeutsche.de/wissen/coronavirus-mutante-b117-daten-1.5197700,
05.02.2021.
[9] RKI: 2. Bericht zu Virusvarianten von SARS–CoV–2 in Deutsch-
land, insbesondere zur Variant of Concern (VOC) B.1.1.7
www.rki.de/DE/Content/InfAZ/N/Neuartiges_Coronavirus/DESH/Bericht_VOC_2021-02-17.pdf?__b
17.02.2020.
[10] COVID–19 Datenhub; time series of daily
COVID–19 infections as reported by RKI,
npgeo-corona-npgeo-de.hub.arcgis.com/datasets/6d78eb3b86ad4466a8e264aa2e32a2e4_0;
access on 17.February.
[11] RKI: Daily Situation Reports, www.rki.de/DE/Content/InfAZ/N/Neuartiges_Coronavirus/Situations
9England
0.8 Denmark
Portugal
Netherlands
0.7 Germany
R_2-R_1 (estimated)
0.6
0.5
0.4
0.3
0.2
2 4 6 8 10 12 14
Week after 1%100
SIR-model
linearized model
RKI Data
80
60
Variant 2 in %
2.5
35
40 Strain
1
=0.71
Strain 2
=0.75
1
30 =0.8
2 20 1
25
0
1.5 0.0 2.5 5.0 7.5 10.0 12.5 15.0 17.5 20.0
dimensionless time
T
20
1
15
0.5
10
0
5
1.25
0 51.3 10 1.35 15 1.4 20 1.45
25 30
1.5
Time
r tYou can also read
